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A Fourth Order Energy Dissipative Scheme for a Traffic Flow Model

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  • Xiaowei Chen

    (College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China)

  • Mingzhan Song

    (College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China)

  • Songhe Song

    (College of Liberal Arts and Sciences, National University of Defense Technology, Changsha 410073, China
    State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, China)

Abstract

We propose, analyze and numerically validate a new energy dissipative scheme for the Ginzburg–Landau equation by using the invariant energy quadratization approach. First, the Ginzburg–Landau equation is transformed into an equivalent formulation which possesses the quadratic energy dissipation law. After the space-discretization of the Fourier pseudo-spectral method, the semi-discrete system is proved to be energy dissipative. Using diagonally implicit Runge–Kutta scheme, the semi-discrete system is integrated in the time direction. Then the presented full-discrete scheme preserves the energy dissipation, which is beneficial to the numerical stability in long-time simulations. Several numerical experiments are provided to illustrate the effectiveness of the proposed scheme and verify the theoretical analysis.

Suggested Citation

  • Xiaowei Chen & Mingzhan Song & Songhe Song, 2020. "A Fourth Order Energy Dissipative Scheme for a Traffic Flow Model," Mathematics, MDPI, vol. 8(8), pages 1-10, July.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1238-:d:390679
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    References listed on IDEAS

    as
    1. Nagatani, Takashi, 1998. "Time-dependent Ginzburg–Landau equation for the jamming transition in traffic flow," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 258(1), pages 237-242.
    2. Akman Yıldız, Tuğba & Uzunca, Murat & Karasözen, Bülent, 2019. "Structure preserving reduced order modeling for gradient systems," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 194-209.
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    Cited by:

    1. Hyun Geun Lee, 2020. "An Efficient and Accurate Method for the Conservative Swift–Hohenberg Equation and Its Numerical Implementation," Mathematics, MDPI, vol. 8(9), pages 1-10, September.
    2. Hyun Geun Lee, 2022. "A Linear, Second-Order, and Unconditionally Energy-Stable Method for the L 2 -Gradient Flow-Based Phase-Field Crystal Equation," Mathematics, MDPI, vol. 10(4), pages 1-9, February.

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